Counting occurrences of some subword patterns

We find generating functions the number of strings (words) containing a specified number of occurrences of certain types of order-isomorphic classes of substrings called subword patterns. In particular, we find generating functions for the number of strings containing a specified number of occurrences of a given 3-letter subword pattern.Comment: 9 page

Counting descents, rises, and levels, with prescribed first element, in words

Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and generalize some of the results in [Counting occurrences of some subword patterns, Discrete Mathematics and Theoretical Computer Science 6 (2003), 001-012.].Comment: 20 pages, sections 3 and 4 are adde

The sigma-sequence and counting occurrences of some patterns, subsequences and subwords

We consider sigma-words, which are words used by Evdokimov in the construction of the sigma-sequence. We then find the number of occurrences of certain patterns and subwords in these words.Comment: 10 page

The Peano curve and counting occurrences of some patterns

We introduce Peano words, which are words corresponding to finite approximations of the Peano space filling curve. We then find the number of occurrences of certain patterns in these words.Comment: 9 pages, 1 figur

Using homological duality in consecutive pattern avoidance

Using the approach suggested in [arXiv:1002.2761] we present below a sufficient condition guaranteeing that two collections of patterns of permutations have the same exponential generating functions for the number of permutations avoiding elements of these collections as consecutive patterns. In short, the coincidence of the latter generating functions is guaranteed by a length-preserving bijection of patterns in these collections which is identical on the overlappings of pairs of patterns where the overlappings are considered as unordered sets. Our proof is based on a direct algorithm for the computation of the inverse generating functions. As an application we present a large class of patterns where this algorithm is fast and, in particular, allows to obtain a linear ordinary differential equation with polynomial coefficients satisfied by the inverse generating function.Comment: 12 pages, 1 figur